Consider This Quotient:${ \left(x^3-8x+6\right) \div \left(x^2-2x+1\right) }$Use Long Division To Rewrite The Quotient In An Equivalent Form As ${ Q(x) + \frac{r(x)}{b(x)} }$, Where { Q(x) $}$ Is The Quotient, [$

Introduction

In algebra, long division is a powerful tool for dividing polynomials. It allows us to rewrite a quotient in an equivalent form, making it easier to work with and understand. In this article, we will explore the process of long division and apply it to the given quotient: $\left(x^3-8x+6\right) \div \left(x^2-2x+1\right)$. We will break down the steps and provide a clear explanation of each, making it easy to follow along.

Understanding the Quotient

Before we begin the long division process, let's take a closer look at the quotient. The quotient is a fraction, where the numerator is a polynomial of degree 3 and the denominator is a polynomial of degree 2. Our goal is to rewrite the quotient in an equivalent form, where the numerator is a polynomial of degree 1 and the denominator is the same as the original denominator.

The Long Division Process

Long division is a step-by-step process that involves dividing the numerator by the denominator. We will divide the leading term of the numerator by the leading term of the denominator, and then multiply the result by the denominator. We will then subtract the product from the numerator and repeat the process until we have a remainder.

Step 1: Divide the Leading Term

The first step in long division is to divide the leading term of the numerator by the leading term of the denominator. In this case, we will divide $x^3$ by $x^2$. This will give us $x$, which is the first term of the quotient.

Step 2: Multiply and Subtract

Next, we will multiply the result from step 1 by the denominator. In this case, we will multiply $x$ by $x^2-2x+1$. This will give us $x^3-2x^2+x$. We will then subtract this product from the numerator.

Step 3: Bring Down the Next Term

After subtracting the product from the numerator, we will bring down the next term. In this case, we will bring down the $-8x$ term.

Step 4: Repeat the Process

We will repeat the process of dividing the leading term of the new numerator by the leading term of the denominator, multiplying the result by the denominator, and subtracting the product from the numerator. We will continue this process until we have a remainder.

Applying Long Division to the Quotient

Now that we have a clear understanding of the long division process, let's apply it to the given quotient: $\left(x^3-8x+6\right) \div \left(x^2-2x+1\right)$. We will follow the steps outlined above and perform the long division.

Step 1: Divide the Leading Term

We will divide $x^3$ by $x^2$. This will give us $x$, which is the first term of the quotient.

Step 2: Multiply and Subtract

Next, we will multiply $x$ by $x^2-2x+1$. This will give us $x^3-2x^2+x$. We will then subtract this product from the numerator.

Step 3: Bring Down the Next Term

After subtracting the product from the numerator, we will bring down the next term. In this case, we will bring down the $-8x$ term.

Step 4: Repeat the Process

We will repeat the process of dividing the leading term of the new numerator by the leading term of the denominator, multiplying the result by the denominator, and subtracting the product from the numerator. We will continue this process until we have a remainder.

The Result of Long Division

After performing the long division, we will have a quotient in the form of $q(x) + \frac{r(x)}{b(x)}$. In this case, the quotient will be $x + \frac{-2x+11}{x^2-2x+1}$.

Conclusion

In this article, we have explored the process of long division and applied it to the given quotient: $\left(x^3-8x+6\right) \div \left(x^2-2x+1\right)$. We have broken down the steps and provided a clear explanation of each, making it easy to follow along. We have also seen the result of long division, which is a quotient in the form of $q(x) + \frac{r(x)}{b(x)}$. This form is useful for simplifying and working with quotients, making it an essential tool in algebra.

Final Answer

The final answer is: $\boxed{x + \frac{-2x+11}{x^2-2x+1}}$

Introduction

In our previous article, we explored the process of long division and applied it to the given quotient: $\left(x^3-8x+6\right) \div \left(x^2-2x+1\right)$. We have seen the result of long division, which is a quotient in the form of $q(x) + \frac{r(x)}{b(x)}$. In this article, we will answer some frequently asked questions about dividing polynomials with long division.

Q: What is the purpose of long division in algebra?

A: The purpose of long division in algebra is to divide a polynomial by another polynomial, resulting in a quotient and a remainder. This process is useful for simplifying and working with quotients, making it an essential tool in algebra.

Q: How do I know when to stop dividing?

A: You will know when to stop dividing when the degree of the remainder is less than the degree of the divisor. In other words, if the remainder is a polynomial of degree 1 or less, you can stop dividing.

Q: What is the difference between a quotient and a remainder?

A: A quotient is the result of dividing a polynomial by another polynomial, while a remainder is the amount left over after the division. In the form $q(x) + \frac{r(x)}{b(x)}$, the quotient is $q(x)$ and the remainder is $r(x)$.

Q: Can I use long division to divide a polynomial by a non-polynomial?

A: No, you cannot use long division to divide a polynomial by a non-polynomial. Long division is only applicable to dividing polynomials by other polynomials.

Q: How do I handle negative coefficients in long division?

A: When handling negative coefficients in long division, you can simply multiply the entire divisor by -1 to make the coefficients positive. This will not affect the result of the division.

Q: Can I use long division to divide a polynomial by a polynomial with a variable in the denominator?

A: No, you cannot use long division to divide a polynomial by a polynomial with a variable in the denominator. This is because the divisor would not be a polynomial, but rather a rational expression.

Q: How do I check my work in long division?

A: To check your work in long division, you can multiply the quotient by the divisor and add the remainder. If the result is the original polynomial, then your work is correct.

Q: Can I use long division to divide a polynomial by a polynomial with a zero coefficient?

A: Yes, you can use long division to divide a polynomial by a polynomial with a zero coefficient. In this case, the quotient will be a polynomial with a zero coefficient, and the remainder will be the original polynomial.

Conclusion

In this article, we have answered some frequently asked questions about dividing polynomials with long division. We have seen the purpose of long division, how to know when to stop dividing, and how to handle negative coefficients and non-polynomial divisors. We have also seen how to check your work in long division and how to handle polynomials with zero coefficients. By understanding these concepts, you will be able to apply long division to a wide range of problems in algebra.

Final Answer

The final answer is: There is no final numerical answer to this article, as it is a Q&A article.